BIOSTATISTICS -III

BIOSTATISTICS -IIIGENERALIZATION OF RESULTS OF A

SAMPLE OVER POPULATION

RECAPRECAP• Types of data, variables , and scales of

measurement • Types of distribution of data , the concept of

normal distribution curve and skewed curves • Measures of central tendency (mean, median,

mode)• Measures of data dispersion or variability,

concept of variance and standard deviation, standard normal curve with standard deviation

STANDARD ERROR-DEFINITIONSTANDARD ERROR-DEFINITION

• Standard error is the measure of extent to which the sample mean deviates from true population mean.

– It helps in determining the confidence limits within which the actual parameters of population of interest are expected to lie

– It is used as a tool in tests of hypothesis or tests of significance.

STANDARD ERROR-CONCEPTSTANDARD ERROR-CONCEPT

• Estimation of population parameters from results/ statistics of sample mean involves two factors– Standard deviation of the population of interest &– Sample size

• The relationship of population standard deviation to sample size is

STANDARD ERROR (SE)STANDARD ERROR (SE) SE= SD/SE= SD/√n√n

FORMULAE FOR ESTIMATION OF FORMULAE FOR ESTIMATION OF STANDARD ERROR(SE) OF SAMPLESTANDARD ERROR(SE) OF SAMPLE

• 1. SE of sample mean= SD/ √n

• 2. SE of sample proportion(p) = √pq/n

• 3. SE of difference between two means[SE(d)]=√SD1/ n1+ SD2/n2

• 4. SE of difference between two proportions= √p1q1/n1+ p2q2/n2

SE/ SEM SE/ SEM (standard error of mean)

• SE is inversely related to square root of sample size ( the larger the sample ,closer the sample mean to population true mean)

• Z scores can be calculated in terms of standard error by which a sample mean lies above or below a population mean

• Z = x - µ / σ

REFERENCE RANGESREFERENCE RANGES

• The 95% limits( REFER TO 2 Std deviations on either side of mean) and are referred to as

REFERENCE RANGE

• For many biological variables they define what is regarded as the

NORMAL RANGE OF THE NORMAL DISTRIBUTION

CONFIDENCE INTERVALCONFIDENCE INTERVAL

• As standard error(the relation between the relation between sample size and population standard sample size and population standard deviation)deviation) is used for estimation of population mean µ, formula is

µ = X ± 2 SE• the variation in distribution of the sample the variation in distribution of the sample

means can also be quantified in terms of means can also be quantified in terms of MULTIPLES OF STANDARD ERROR(SE)MULTIPLES OF STANDARD ERROR(SE)

Conventionally!!!!!!!!

• 1.96 /2 SE 1.96 /2 SE on either side of mean is taken as the limit of variability.

• These values are taken as CONFIDENCE CONFIDENCE LIMITS LIMITS with intervening difference being

THE 95% CONFIDENCE INTERVAL THE 95% CONFIDENCE INTERVAL whichGives an estimated range of values which is

likely to include an unknown” POPULATION PARAMETER” .

WIDTH OF CONFIDENCE INTERVALWIDTH OF CONFIDENCE INTERVAL

• Reflects how uncertain we are about an unknown parameter

• A wider confidence interval may indicate need for collection of more data before commenting on the population parameter

Reference range vs Reference range vs confidence intervalconfidence interval

• Reference range refers to individuals in populations with standard deviations

• Confidence interval refers to standard error in data estimated from samples

Confidence interval for difference Confidence interval for difference between two meansbetween two means

• It specifies the range of values within which the means of the two populations being compared would lie as they are estimated from the respective samples

• If confidence interval includes “ZERO” we say, ““THERE IS NO SIGNIFICANT DIFFERENCE THERE IS NO SIGNIFICANT DIFFERENCE

BETWEEN THE MEANS OF THE TWO BETWEEN THE MEANS OF THE TWO POPULATIONS AT A GIVEN LEVEL OF POPULATIONS AT A GIVEN LEVEL OF CONFIDENCECONFIDENCE

THE 95 % CONFIDENCE INTERVALTHE 95 % CONFIDENCE INTERVAL• Means we are 95% sure or confident that the

estimated interval in sample contains the true difference between the two population means

(the basic concept remains one of capturing 95% the basic concept remains one of capturing 95% of data within 2 standard deviations of the of data within 2 standard deviations of the standard normal curve of distribution of data standard normal curve of distribution of data in nature) in nature)

• Alternately, 95% of all confidence intervals estimated in this manner (by repeated sampling ) will include the true difference

Practice and clarification time!!!!Practice and clarification time!!!!

Sample of 100 women , Hb 12 gmstandard deviation( 0- 2gm)

• µ= X ± 2 SE OR X ± 2 SD/√N• µ (ci)= 12±[ 2x 2/√100

•=12±[4/10or0.4]• µ (ci)= 12± 0.4•=11.6- 12.4• INTERPRET ????

ROLE OF SAMPLE SIZE AND SD

• µ= X ± 2 SE OR X ± 2 SD/√N• µ (ci)= 12±[ 2x 2/√9• =12±[4/3or 1.33]• µ (ci)= 12±1.33• =10.66- 13.33

• INTERPRET ????

LARGER SD OF 4 GM% ?

• µ= X ± 2 SE OR X ± 2 SD/√N• µ (ci)= 12±[ 2x4/√9• =12±[8/3or 2.66]• µ (ci)= 12±2.66• =9.33- 14.66

• INTERPRET ????

SMALLER SD 0F 0.5 GM Hb

• µ= X ± 2 SE OR X ± 2 SD/√N• µ (ci)= 12±[ 2x0.5/√9• =12±[1/3or 0.33]• µ (ci)= 12±0.33• =11.6- 12.33

• INTERPRET ????

Comment about sample authenticity if true population mean is known(11.2gm)• µ= X ± 2 SE OR X ± 2 SD/√N• µ (ci)= 12±[ 2x 2/√100• =12±[4/10or0.4]• µ (ci)= 12± 0.4• =11.6- 12.4• What about sample mean’s predictive value• ?????? Representative of population under

study or not?????????

Difference of proportionDifference of proportion5200 workers in total population of 10000,(52%) sample of 100 individuals with 0.4 or 40% workers

• What is the possible range of workers we expect to find in the sample of 100 with 95% confidence?

• What conclusions/comments will be drawn about authenticity of sample under consideration?

Standard error of proportionStandard error of proportionp= probability of being workerq= probability of being non worker• P(in pop)= 52% q(in pop)= 48% !!!!!!• SE for proportion= √pq/n=

√52x48/100=√25=5 • P (CI)= p ± 2 SE = 52± 2 x5 = • 42% -62%• { sample’s proportion of workers = 40%}

• COMMENT ????????????????

difference between two proportionsdifference between two proportions

• Proportion of measles infection after vaccination with vacc A(p1) = 22/90=0.244(24.4%)

q1= 100-2.44= 75.6%• Proportion of measles infection after

vaccination with vacc B (p2) = 14/86 = 0.162(16.2%) q2= 100- 16.2= 83.3%

Difference p1-p2= 24.4-16.2= 8.2

Standard error of difference between Standard error of difference between two proportionstwo proportions

• SE =√p1q1/n1 +p2q2/n2= √24.4x75.6/90+16.2x83.8/86

= √ 20.79 +15.76 = √ 36.27 = 6Difference p1-p2= 24.4-16.2= 8.2FOR CI REMEMBER 2±SE SO SE= 4- 8 ( what about 8.2????)

COMMENT !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

THANK YOU FOR APPRECIATING

LOGIC OF BIOSTATISTICS